THE DICHOTOMY PROPERTY IN STABILIZABILITY OF 2 × 2 LINEAR HYPERBOLIC SYSTEMS

. This paper is devoted to discuss the stabilizability of a class of 2 × 2 non-homogeneous hyperbolic systems. Motivated by the example in [5, Page 197], we analyze the influence of the interval length L on stabilizability of the system. By spectral analysis, we prove that either the system is stabilizable for all L > 0 or it possesses the dichotomy property: there exists a critical length L c > 0 such that the system is stabilizable for L ∈ (0 , L c ) but unstabilizable for L ∈ [ L c , + ∞ ). In addition, for L ∈ [ L c , + ∞ ), we obtain that the system can reach equilibrium state in finite time by backstepping control combined with observer. Finally, we also provide some numerical simulations to confirm our developed analytical criteria


Introduction and Main Result
Hyperbolic systems play a crucial role in representing physical phenomena and possess both theoretical and practical significance.Extensive research has focused on well-posedness and control problems, including the stability and stabilization of these systems.In particular, researchers have studied the exponential stability or stabilization of hyperbolic systems without source terms in both linear and nonlinear cases, under various boundary controls such as Proportional-Integral control and backstepping control [21].Previous works by [6,9,10,12,23,27] have also investigated this topic.However, most physical equations, such as the Saint-Venant equations (see Chapter 5 of [8]), Euler equations (see [15] or [20]), and Telegrapher equations, cannot neglect the source term.Therefore, it is crucial to investigate the dynamics of hyperbolic systems with source terms.
Two main approaches have been used to achieve asymptotic stability of hyperbolic systems: analyzing the evolution of the solution along characteristic curves, as extensively studied in previous works such as [1,7,22,25,26]; and relying on a Lyapunov function approach, as thoroughly researched in [8-11, 14, 16, 18, 28, 29].Both of these approaches are concerned with obtaining stability for the system.
Another strategy that has been studied is the Backstepping method, which aims to design a control law that achieves stabilization.The Backstepping method has been applied in [13,21] and typically requires full-state feedback control.However, it is possible to achieve boundary state feedback through backstepping control by designing an appropriate observer, as demonstrated in [2,3].
In [19], Gugat and Gerster analyze the limit of stabilizability for a network of hyperbolic systems.Remarkably, their results reveal that under certain conditions, the system may be inherently unstabilizable.
These results inspire us to investigate whether the stabilizability of the system (1.1) possesses the dichotomy property on L. Here, the dichotomy property on L can be described as follows: there exists a critical value L c > 0 such that: • While L ≥ L c , the system is not stabilizable, i.e. the system cannot be exponentially stable for any discussed control.
• While 0 < L < L c , the system is stabilizable, i.e. there exists certain control such that the system is exponentially stable.
In this paper we discuss the boundary feedback stabilization of the following 2 × 2 linear hyperbolic systems over a bounded interval [0, L]: where λ > 0 and a, b ∈ R are given constants, y 0 1 , y 0 2 ∈ L 2 (0, L) are the initial data.The boundary feedback law takes the proportional form u(t) = ky 2 (t, 0), (1.3) where k ∈ R is the tuning parameter and y 2 (t, 0) is the output measurement.
We are concerned about the exponential stability of the closed-loop system (1.2).
Definition 1.1.The linear hyperbolic system (1.2) (1.3) is said to be L 2 exponentially stable if there exists C > 0 and α > 0 such that, for every (y 3) satisfies: In this paper, we propose a method for finding the critical value L c , given fixed parameters a, b, λ.
Our approach is based on spectral analysis.For values of |k| ≥ 1, we demonstrate that the closed-loop system (1.2) (1.3) is not exponentially stable by identifying unstable eigenvalues, specifically those located on the right half of the complex plane.To achieve this, we approximate the characteristic function at infinity and use Rouché's Theorem to obtain the roots.In the case of |k| < 1, we introduce the function N a,b,λ (k, L) to represent the degree of the characteristic function, see (3.16), on the right side of the complex plane.As stated in Lemma 3.2, N a,b,λ remains constant within each block that is separated by marginal curves determined by A a,b,λ .Furthermore, N a,b,λ ≡ 0 within the block at the bottom.By applying Lemma 3.3, we show that N a,b,λ increases by 1 when (k, L) moves from one block to another block above it.Therefore, we conclude that the stability region is the block at the bottom.Finally, we determine the critical value L c using the analytical results obtained from the marginal curves determined by A a,b,λ .
Theorem 1.1.Let λ > 0 be fixed.Then, either the system (1.2) (1.3) is stabilizable for all L > 0 or it possesses the dichotomy property.More precisely, the expression of L c in terms of a, b ∈ R is explicitly given as follows (see Figure 1.1) : +∞, if else. (1.5) Here L c := +∞ means that the system is stabilizable for all L > 0.
Remark 1.2.While L is sufficiently small, we can establish a Lyapunov function to demonstrate that the system (1.2) (1.3) is exponentially stable.For instance, if ab > 0, |k| < ε < 1, we define ) is a Lyapunov function and the corresponding system is exponentially stable.
Remark 1.3.The general hyperbolic system with the rightward speed λ 1 > 0 and leftward speed (1.6) can be reduced, through a scaling of the space variable x → λ 1 x, to a system with rightward speed 1 and leftward speed λ 1 > 0 in the form of (1.2).Thus, Theorem 1.1 can be extended to the general system (1.6).
According to Theorem 1.1, the proportional feedback control (1.3) cannot stabilize the system (1.2) for L > L c .Therefore, an alternative control approach is worth exploring in this case.Building upon the works of Hu et al. [13,21] and Holta et al. [2], we develop a Backstepping control combined with observer design that stabilizes the system even when L ≥ L c , and without the need to observe the full state.Notably, the proposed control law drives the system to its zero equilibrium in finite time.
More details are presented in Section 5.
The main contribution of this paper can be summarized in three aspects: 1) we provide a complete characterization of the stabilizability of the hyperbolic system (1.2) under proportional feedback control (1.3) for all cases; 2) we show that the stabilizability of the system exhibits a dichotomy property on the interval L, indicating a clear boundary between the stabilizable and non-stabilizable regions; 3) we propose a new control method that combines backstepping control with observer design to stabilize the system when the proportional control fails.
The organization of this paper is as follows.In Section 2, we provide some preliminaries including Spectral Mapping Property and Implicit Function Theorem, which will be used in the following Sections.In Section 3, we provide the proof of Theorem 1.1.In Section 4, we provide some numerical simulations to confirm our developed analytical criteria in Section 3. In Section 5, We give a sketch of the construction of the Backstepping control with observer design for the case of L ≥ L c .
Notations.In this paper, we use standard notation and terminology in complex analysis and algebra.Specifically, C+, C−, and C 0 denote the sets of complex numbers with positive real parts, negative real parts, and zero real parts, respectively.We use C+ to denote the set

Preliminaries
Applying the results in Lichtner [24], we have the following lemmas: Lemma 2.1.Let S(t)(t ≥ 0) be the C 0 semigroup on L 2 (0, L) that corresponds to the solution map of (1.2) (1.3), and let A be the infinitesimal generator of the semigroup S(t)(t ≥ 0).Let us denote by σ p (A ) and σ(A ) the point spectrum and the spectrum of A , respectively.Then, σ(A ) = σ p (A ).
Moreover, A has the Spectral Mapping Property (SMP), that is Hence (SMP) contains spectrum determined growth From Lemma 2.1, we obtain the following proposition: Proposition 2.1.The system (1.2) (1.3) is not exponentially stable if and only if s(A ) ≥ 0.
We will apply the analytic implicit function theorem in the proof of Lemma 3.3.The Implicit Function Theorem from [17] is stated following: and (z 0 , w 0 ) ∈ B a point with f (z 0 , w 0 ) = 0 and Then there is an open neighborhood U = U ′ × U ′′ ⊂ B and a holomorphic map g :

Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1.We first establish the characteristic equation.
Note that cosh(ηL) and sinh(ηL) η are analytical functions with respect to η 2 , which further implies that the left side of Eq. (3.10) is analytic with respect to σ. Denote by with and The set of the roots of F a,b,λ,k,L (σ) satisfy By Lemma 2.1, we obtain that a sufficient and necessary condition for the stability of the system (1.2) (1.3): Denote by Our goal is to establish the region for parameter a, b, λ, k, L such that N a,b,λ (k, L) = 0. To acheive this, we fix a, b, λ and figure out the stability region for k, L on R × [0, +∞).We divide into two case Proof of Lemma 3. The solutions of Eq.(3.18) include Considering the solutions on the imaginary axis η = αi(α ̸ = 0, α ∈ R), we obtain which has infinitely solutions.This implies that F a,b,λ,k,L (σ) has infinitely many roots on the imaginary axis.
These analysis for three cases complete the proof of Lemma 3.1.□

|k| < 1
From Lemma 3.1, only we choose |k| < 1 can the system be exponentially stable.We then prove the main theorem by spectral analysis.
Therefore, there exists a sufficently large R > 0 such that Eq.(3.21) has no roots located in σ ∈ C + ∩ {σ||σ| ≥ R}.Then we establish Eq.(3.20).Eq.(3.20) yields  We can further demonstrate that if a point (k, L) moves from one block to a block above it, then N a,b,λ increases by 1.Therefore, for any point (k, L) in a block other than Block I, the system (1.2) (1.3) possesses at least one eigenvalue in C + and cannot be exponentially stable.

Numerical Simulations
In this section we present some numerical simulations generated with MATLAB of upwind scheme with implicit methods for the system (1.2) (1.3).We adopt the finite difference method in both the time and the space domain, which can be written as follows.The grid size N = 100 and the time step ∆t = 2L/N are used.

Backstepping control
In this section, we want to use Backstepping method combined with the observer design to stabilize the system with the case that cannot be stabilized by the proportional feedback control.We first make a scaling of space variable x → L − x, then the control could be on the right side and the boundary condition be on the left.Theorem 1.1 can apply to the following system: ∂ t y 2 + λ∂ x y 2 + by 1 = 0, (t, x) ∈ (0, +∞) × (0, L), y 2 (t, 0) = y 1 (t, 0), t ∈ (0, +∞), (5.1) The output is Applying the results in Anfinsen et al. [2], we design the following observer: with Γ 1 (x), Γ 2 (x) are injection gains to be designed.
Thus we get the main theorem in this section.We now divide into three cases: Case I. ab = 0, the corresponding η = 0. Eq.(A.4) yields:

1 .
The sets of integers, positive integers, and non-negative integers are denoted by Z, N * , and N, respectively.The imaginary unit is denoted by i such that i = √ −For σ ∈ C, we use Reσ, Imσ, argσ, and |σ| to denote the real part, imaginary part, principal value of argument, and norm of σ, respectively.For an analytic function f , an open subset Ω ⊆ C and b ∈ C, deg(f, Ω, b) denotes the number of roots of the equation f (z) = b, counted by multiplicty.

. 26 )
Since F a,b,λ,k(t),L(t) (σ) ̸ = 0 on C, the right side of Eq.(3.26) is continuous with respect to t.Furthermore, since N a,b,λ (k(t), L(t)) is an integer, we know that N a,b,λ (k(t), L(t)) is a constant due to the discontiuity between two different integers.□As shown in Fig.1, A a,b,λ divides the k − L plane into several blocks.

Fig. 1 .
Fig. 1. k − L plane is seperated by marginal curves determined by A a,b,λ for different cases.

Fig. 2 .
Fig. 2. k − L plane is seperated by marginal curves determined by A a,b,λ for different cases.